He can lay claim to running a larger executive branch than any of his rivals, with the exception of Joseph R. Biden Jr.’s two terms as vice president. (New York City has more residents than the home states of each of the three governors in the race, Colorado, Washington and Montana.) And he can tout New York’s booming economy and falling crime rate while pointing to a flurry of other liberal agenda items he has pursued.

Mr. de Blasio, 58, rose to the mayoralty as a self-styled progressive, and he has repeatedly called for the national party to move to the left. Back home, though, he has proved a more cautious politician. He has supported incumbent Democratic politicians against progressive insurgents, for instance, and he has drawn persistent criticism from his left flank. An early biography of Mr. de Blasio was titled “The Pragmatist.”

As a bearded young man in the late 1980s, Mr. de Blasio admired the cause of the leftist Sandinistas of Nicaragua. But in New York, he rose to power as a political insider who worked in City Hall and later managed Hillary Clinton’s 2000 Senate campaign before starting his own political career.

Here are some of the issues he has advocated, and confronted, as mayor.

Income inequality

The gap between the rich and the poor, and the lifestyles they can afford, has been an animating issue for Mr. de Blasio for more than a decade, and a central message of his mayoral campaigns. He was an adviser to and supporter of John Edwards’s presidential campaign in 2004, drawn to the former senator’s “two Americas” pitch about the troubling stratification of wealth in the country.

But as mayor, Mr. de Blasio has been unsuccessful in seeking a new “millionaires’ tax” in New York City. The trouble is that the state, not the city, holds the power to impose taxes in New York, and Mr. de Blasio has been unable to convince state legislators or the governor to sign onto his various tax plans for high earners.

The line has fast become a favorite for a mayor who, in 2015, tried to organize a forum in Iowa on income inequality. The idea of a forum collapsed when neither Mrs. Clinton nor Senator Bernie Sanders of Vermont agreed to attend.

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Mr. de Blasio’s signature policy achievement in New York is his implementation of a universal prekindergarten program.CreditMichael Nagle for The New York Times

Education

Mr. de Blasio’s signature policy achievement in New York is his implementation of a universal prekindergarten program. There are about 70,000 children now enrolled in prekindergarten citywide, up from 19,000 in 2013, the year he was first elected.

The program scaled up quickly — and successfully. Last year a prominent national evaluation system rated about 94 percent of the city’s pre-K programs at a threshold that would predict positive long-term outcomes for students.

The added free year of schooling has been widely popular, and Mr. de Blasio has already begun to expand a follow-up program for 3-year-olds, commonly known as 3-K.

Immigration

The mayor pushed to make New York City a so-called sanctuary city — limiting cooperation with federal immigration authorities — even before President Trump took office. He also supported the creation of a city ID card that immigrants can use for both proof of residency and access to some municipal programs and facilities.

This year, Mr. de Blasio announced a plan to spend $100 million to ensure that undocumented immigrants and others who cannot qualify for insurance can receive medical treatment.

But immigration advocates criticized Mr. de Blasio for a proposal in 2017 to no longer pay for legal representation for people facing deportation if they had been convicted of certain crimes in the past.

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Crime in New York City has plunged in recent years.CreditRyan Christopher Jones for The New York Times

Criminal justice

Crime in New York City has plunged in recent years, with violent crimes such as murders and shootings dropping to new lows. The declines came even after Mr. de Blasio continued to scale back the city’s so-called stop-and-frisk program, a decision that critics had warned would lead to a spike in crime.

The program, in which police officers stopped and searched thousands of people on the streets each year, had been plagued by racial disparities. While those issues remain, the total number of street stops has fallen 98 percent since 2011.

Now, Mr. de Blasio likes to hail New York as “the safest big city in America.”

Mr. de Blasio’s stance on the legalization of recreational marijuana, which has been championed by some of the other 2020 Democratic contenders, has been more hesitant. He was long opposed but shifted his position late last year, backing “a regulatory framework” for legalization days after the Democratic governor, Andrew M. Cuomo, endorsed recreational marijuana.

Housing

Mr. de Blasio has made affordable housing in the city a major theme of his administration, and has announced plans to preserve or build 300,000 affordable housing units by 2026. He created or preserved a record 24,500 units in 2017 alone, spending $1.1 billion.

But housing prices have continued to spiral upward in the city, and homelessness has grown. The New York City Housing Authority, home to more than 400,000 low-income New Yorkers, has been beset with problems that predate Mr. de Blasio and his administration, though he has been criticized as slow to respond to certain issues, such as toxic lead poisoning. The city recently entered into a settlement that gave the federal government greater control of public housing.

A version of this article appears in print on , on Page A24 of the New York edition with the headline: Where de Blasio, Professed Progressive Mayor, Stands on City’s Top Issues. Order Reprints | Today’s Paper | Subscribe